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2.3. THE PRODUCT AND QUOTIENT RULES
The quotient rule
Because quotients and products are closely linked, we can use the product rule to understand how to take the derivative of a quotient. In particular, let Q(x) be defined by
Q(x) = f (x)/g(x), where f and g are both differentiable functions. We desire a formula
for Q ′ in terms of f , g, f ′ , and g ′ . It turns out that Q is differentiable everywhere that
g(x) 0. Moreover, taking the formula Q = f /g and multiplying both sides by g, we can
observe that
f (x) = Q(x) · g(x).
Thus, we can use the product rule to differentiate f . Doing so,
f
′ (x) = Q(x)g
′ (x) + g(x)Q
′ (x).
Since we want to know a formula for Q ′ , we work to solve this most recent equation for
Q ′ (x), finding first that
Q
′ (x)g(x) = f
′ (x) − Q(x)g
′ (x).
Dividing both sides by g(x), we have
Q
′ (x) =
f ′ (x) − Q(x)g ′ (x)
g(x)
.
Finally, we also recall that Q(x) =
f (x)
g(x) . Using this expression in the preceding equation
and simplifying, we have
Q
′ (x) =
f ′ (x) −
f (x)
g(x) g ′ (x)
g(x)
=
f ′ (x) −
f (x)
g(x) g ′ (x)
g(x)
·
g(x)
g(x)
=
g(x) f ′ (x) − f (x)g ′ (x)
g(x) 2
.
This shows the fundamental argument for why the quotient rule holds.
Quotient Rule: If f and g are differentiable functions, then their quotient Q(x) =
f (x)
g(x)
is also a differentiable function for all x where g(x) 0, and
Q
′ (x) =
g(x) f ′ (x) − f (x)g ′ (x)
g(x) 2
.
Like the product rule, it can be helpful to think of the quotient rule verbally. If a
function Q is the quotient of a top function f and a bottom function g, then Q ′ is given
by “the bottom times the derivative of the top, minus the top times the derivative of